Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 24 / 3 / b / Solution

Count affine solutions and then add the point at infinity. Over the finite field ${\mathbb{F}}_2$, both values of $x$ give $x^3-x=0$, and both values of $y$ solve $y^2+y=0$. Hence there are four affine points. Over ${\mathbb{F}}_3$, all three values of $x$ give zero, and each has the two ordinates $0,2$, giving six affine points.

Over the finite field ${\mathbb{F}}_{11}$, completing the square gives $(2y+1{)}^2=1+4(x^3-x)$. The square residues are $0,1,3,4,5,9$. For $x=0,1,\dots ,10$, the right sides and the corresponding numbers of ordinates areTheir sum is sixteen. The resulting elliptic-curve point counts over a finite field are